Read ALL of this first, then proceed:
If you are upgrading, you shouldn't have to do a thing. Mine worked fine going through an upgrade to Win 10. If you're doing a clean install (which I've also done moving to an new SSD,) do this:
1. MAKE A BACKUP of these two folders:
C:\Users\<your user name here>\AppData\Local\Activision. Note that this folder could be quite large as it also contains all the maps you've played.
C:\Program Files (x86)\Activision
2. Build your new PC
3. If you have a CD goto step 4a. If installed from Steam go to step 5
4a. Install COD5 from the original CD. Don't bother starting the game at this point.
4b. Apply all COD5 patches in order. The patches are available in our download area.
4c. Copy the backup you made of C:\Program Files (x86)\Activision right over the new installation. This should correct any PunkBuster update issues since your old version was probably working fine.
4d. Skip to step 6
5. Install from Steam. I presume it already comes pre-patched so you shouldn't have to deal with that.
6. COPY the backup of your AppData\Local\Activision folder to your new c:\users\<username>\AppData\Local folder. (COPY not MOVE; you want a safe copy of this folder always!)
7. Start the game, and have fun. You should be right where you left off.
8. If you went through Step 4 and the game will not run, try putting in the CD and running d:\DirectX\DXSETUP.exe. This is the DX9 runtime needed for the game. If you're having problems an you installed from Steam, check their forums for answers, or maybe someone else here can guide you. Alternatively, just look for a DX9 installer that will run on Windows 10 from Microsoft. DON'T download DirectX drivers from anywhere but Microsoft. Can't stress that last statement enough.
9. Once you're sure everything is okay, then, and only then, is it safe to delete the copy of your original AppData\Local\Activision folder. If you want to maintain a lighter version as a safeguard, you can delete the AppData\Local\Activision\CoDWaW\usermaps folder from your backup. That will lighten the load by many GB.
Hope that helps.